I have a classic racing bike, the wheels on the bike are not quick release system. I am thinking of putting quick release wheels on the bike, and looking for a descent set. Is it possible to convert the existing wheels to quick release wheels?

Possible but it depends on the existing hubs and if you can find the parts to do the conversion. Low end alloy or steel hubs would probably not be worth the trouble and expense.

I've done it with ubiquitous Normandy high-flange hubs, but can't say what other brands are interchangeable.

Front hub and fork are the first areas of review. Many front solid axle hubs have a smaller diameter axle, to fit the larger diameter axle of a quick release hub, the fork would need to be modified to accept the axle. All can be done, time, tools and talent.

For the rear wheel, the overall width of many solid axle wheels locknut to locknut face is 120mm, pre built wheels almost universally now are wider. The frame would need to be sprung wider upon every install or frame re spaced.

It can all be done.

I've converted the rear axle of a couple of bikes to QR (< 10 years old). It was a bit of a pain. Inevitably the axle sizes or threads are different so one can't just move the cones and spacers directly from the old axle to the new axle. So you have to find a new axle, and matching cones, spacers, and lock nuts. And if the dust seals are integrated to the cones, they may need matched too, or moved.

Inevitably one gets the rear spacing off just slightly, and needs to adjust the dishing of the wheel.

A cheap matching donor hub may be of benefit.

You could probably get axles and cones from Wheels Manufacturing but you mighf want to try fitting some from another bike guy first. You would want to verify sizes.

I've done this twice, and it was more complicated than I guessed it would be. Lots of different sizes on cones and axels. Luckily I have a big pile of surplus wheels to dig through and found all the parts except for a couple cones (to replace pitted ones) which my LBS had on hand.